Congruent figures

Two figures are congruent if you can move one (slide, rotate, flip) to exactly cover the other. Their corresponding parts , sides and angles , must match.

Idea

We write congruence with the symbol $\cong$. For two triangles, $\triangle ABC \cong \triangle DEF$ means:

• $AB = DE$, $BC = EF$, $CA = FD$ (sides match in pairs).
• $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$ (angles match in pairs).

The order of letters matters. Writing $\triangle ABC \cong \triangle DEF$ specifically says $A$ corresponds to $D$, $B$ to $E$, $C$ to $F$. So $AB$ matches $DE$, $\angle A$ matches $\angle D$, and so on. Pick the order carefully.

Congruence is a stronger condition than similarity (which only requires same shape, not same size). All squares are similar; but only squares of the same side are congruent.

Examples of congruent figures around us: two pages of the same book, two copies of the same coin, two factory-made bricks. Differences in real life (small imperfections) are ignored in geometry.

The everyday word "identical" captures the idea of congruence well.

Worked examples

Example 1. Two squares have side $5$ cm each. Are they congruent? Why?

Yes , same shape (square) and same size (side $5$). Each pair of corresponding sides is equal; each angle is $90^\circ$.

Example 2. A rectangle has sides $3$ and $4$; another has sides $4$ and $3$. Are they congruent?

Yes , same shape, same dimensions (just labelled differently). One can be rotated by $90^\circ$ onto the other.

Example 3. If $\triangle ABC \cong \triangle PQR$, list all corresponding sides and angles.

$AB = PQ$, $BC = QR$, $CA = RP$. $\angle A = \angle P$, $\angle B = \angle Q$, $\angle C = \angle R$.

Example 4. Are two equilateral triangles always congruent?

No , only if they have the same side length. Same shape (equilateral) but possibly different sizes. They are similar always.

Try it yourself

1. Are two circles always congruent? When?
2. If $\triangle ABC \cong \triangle XYZ$ and $AB = 5$, find $XY$.
3. List corresponding parts if $\triangle PQR \cong \triangle LMN$.
4. Are two squares always congruent? When?
5. Two triangles have angles $50^\circ, 60^\circ, 70^\circ$ each. Must they be congruent?
6. State the difference between congruent and similar figures.
7. A factory makes bricks $20$ cm $\times 10$ cm $\times 6$ cm. Are any two bricks congruent?
8. Sketch two triangles that are congruent but in different orientations.

Activity

Cut out two identical paper figures (same triangle, square or any shape). Lay one over the other in different positions , flip, rotate, slide. They always coincide. That is congruence in action.